Nondisk tips

The quantitative SECM theory has been developed for various heterogeneous and homogeneous processes and for different tip and substrate geometries [56-59]. Here, we survey the theory pertinent to an inlaid disk electrode (Fig. 5) approaching a flat substrate, which can be considered infinitely large as compared to the tip size. The case of finite substrate size was treated by Bard et al. [60]. The theory for nondisk tips (e.g., shaped as a cone or a spherical cap) is discussed in Refs. [12-14] and [58], and in Section IV.B.2. 

Nann and Heinze [39] developed and used an adaptive finite element (AFE) algorithm to carry out simulations of steady-state, chronoamperometric, and voltammetric SECM responses for various nondisk tip geometries, for example, hemispherical, conical, and capillary electrodes. 

Fulian et al. [75] introduced the boundary element method (BEM) for the niunerical solution of SECM diffusion problems and showed that it is more suitable for problems with regions of complex or rapidly changing geometries than finite difference methods employed in earlier SECM publications. The BEM was used to simulate current responses for a nondisk tip approaching a flat substrate a disk-shaped tip over a hemispherical or a sphere-cap substrate, or a tilted substrate a lateral scan of a disk-shaped tip over an insulating/conductive boundary. 

Although disk-shaped tips are typically most useful for SECM experiments, it is not always possible to produce such tips, especially when they have to be nanometer sized. For some special applications (e.g., penetration experiments), one may want to purposely fabricate tips with different geometries. To characterize nondisk shaped tips, experimental approach curves were obtained and then compared to simulated ones [12]. A number of UME tip geometries including hemispheres [14, 15], spheres [16], rings [17], ring-disks [18], and etched electrodes [19, 33] have been characterized in this way. 

Another type of nondisk-shaped SECM tips are UMEs shaped as spherical caps. They can be obtained, for example, by reducing mercuric ions on an inlaid Pt disk electrode or simply by dipping a Pt UME into mercury [15]. An approximate procedure developed for conical geometry was also used to model spherical cap tips [12]. Selzer and Mandler performed accurate simulations of hemispherical tips using the alternative direction implicit final difference method to obtain steady-state approach curves and current transients [14]. As with conical electrodes, the feedback magnitude deceases with increasing height of the spherical cap, and it is much lower for a hemispherical tip than for the one shaped as a disk. 

Several advantages of the inlaid disk-shaped tips (e.g., well-defined thin-layer geometry and high feedback at short tip/substrate distances) make them most useful for SECM measurements. However, the preparation of submicrometer-sized disk-shaped tips is difficult, and some applications may require nondisk microprobes [e.g., conical tips are useful for penetrating thin polymer films (18)]. Two aspects of the related theory are the calculation of the current-distance curves for a specific tip geometry and the evaluation of the UME shape. Approximate expressions were obtained for the steady-state current in a thin-layer cell formed by two electrodes, for example, one a plane and the second a cone or hemisphere (19). It was shown that the normalized steady-state, diffusion-limited current, as a function of the normalized separation for thin-layer electrochemical cells, is fairly sensitive to the geometry of the electrodes. However, the thin-layer theory does not describe accurately the steady-state current between a small disk tip and a planar substrate because the tip steady-state current iT,co was not included in the approximate model (19). 

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